Gourmet kitchens. Indoor basketball courts. The newest place for luxury amenities isn’t at your home: It is in your storage unit.

As the top end of the real-estate market boomed in many cities, a parallel market for high-end storage appeared as well. Fort Lauderdale, Fla.’s RoboVault, which offers a biometric fingerprint-access security system and an automated robotic parking section, says revenues are up 80% since 2013. In California, Garage Unlimited of Monterey is targeting car collectors with its 46 units, each with…